May anyone can help in this case.
I have a map F:
in a different note:
I must to find out if this map is linear, surjective, injective?
The first point I already done, so I know the map is linear. I have more problems with second and third point. My thinking:
1) the map is not injective because:
2) but the map is surjective because:
for every g in G exits d in D that is F(d)=g
Are those two arguments correct?
Thanks in advance!
Be careful about the definitions of injective and surjective. As Drexel28 mentioned, this is asking about the injectivity and surjectivity of F (which maps 3 x 3 matrices to other 3 x 3 matrices). Your domain and range are 3 x 3 matrices.
Let be 3 x 3 matrices.
Here's how injectivity and surjectivity should be interpreted in the problem:
The entire time, we are talking about the entire matrix, not the elements inside the matrix.
So if we have two matrices:
So the map is not injective because .
If we change only elements and all the others stay the same we always get the same output matrices for those set. Special case is when we get the null matrix, then must be:
Is that right?
For surjectivity your hint is a very good one, since in this case it's clear what the result is but you could have (if you are aware of it) appealed to the (common?) theorem which says that if is finite dimensional then is a monomorphism (fancy word for injective linear transformation) if and only if it's an epimorphism (fancy word for surjective linear transformation). This is just a comment though.
So if I note this in more formal form:
We take the matrix
In our case the map not allows us to get such output so the form that provides surjectivity does not stand up.
So and the map is not surjective.
So all of upper triangular matrices are not surjective?